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Theorem isoml 37238
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐵 = (Base‘𝐾)
isoml.l = (le‘𝐾)
isoml.j = (join‘𝐾)
isoml.m = (meet‘𝐾)
isoml.o = (oc‘𝐾)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isoml
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6767 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isoml.b . . . 4 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2796 . . 3 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6767 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 isoml.l . . . . . . 7 = (le‘𝐾)
64, 5eqtr4di 2796 . . . . . 6 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5085 . . . . 5 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
8 fveq2 6767 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 isoml.j . . . . . . . 8 = (join‘𝐾)
108, 9eqtr4di 2796 . . . . . . 7 (𝑘 = 𝐾 → (join‘𝑘) = )
11 eqidd 2739 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
12 fveq2 6767 . . . . . . . . 9 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13 isoml.m . . . . . . . . 9 = (meet‘𝐾)
1412, 13eqtr4di 2796 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = )
15 eqidd 2739 . . . . . . . 8 (𝑘 = 𝐾𝑦 = 𝑦)
16 fveq2 6767 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 isoml.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17eqtr4di 2796 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
1918fveq1d 6769 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( 𝑥))
2014, 15, 19oveq123d 7289 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ( 𝑥)))
2110, 11, 20oveq123d 7289 . . . . . 6 (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 (𝑦 ( 𝑥))))
2221eqeq2d 2749 . . . . 5 (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 (𝑦 ( 𝑥)))))
237, 22imbi12d 345 . . . 4 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
243, 23raleqbidv 3334 . . 3 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
253, 24raleqbidv 3334 . 2 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
26 df-oml 37179 . 2 OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
2725, 26elrab2 3627 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064   class class class wbr 5074  cfv 6427  (class class class)co 7268  Basecbs 16900  lecple 16957  occoc 16958  joincjn 18017  meetcmee 18018  OLcol 37174  OMLcoml 37175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-iota 6385  df-fv 6435  df-ov 7271  df-oml 37179
This theorem is referenced by:  isomliN  37239  omlol  37240  omllaw  37243
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